Journal of Combinatorial Theory, Series A, 54 (1990), pp. 272-295.
Ömer Egecioglu and Jeffrey B. Remmel
Monomial Symmetric Functions and the Frobenius Map
Abstract.
There is a well known isometry between the center $Z(S_n)$ of the group algebra of the
Symmetric group $S_n$ and the space of homogeneous symmetric functions $H^n$ of degree
$n$. This isometry is defined via the Frobenius map $F:Z(S_n) -> H^n$, where
$F(f) = \frac{1}{n!} \sum_{\sigma \in S_n} f(\sigma )\psi_{\lambda (\sigma )}.$
Let $M^\lambda = F^{-1} (m_\lambda )$ be the preimage of the monomial symmetric
function $m_\lambda$ under $F$. We give an interpretation of $M^\lambda$ in terms
of certain combinatorial structures called $\lambda$-domino tabloids. Using this
interpretation, a number of properties of $M^\lambda$ can be derived. The
combinatorial interpretation of the preimage of the so called forgotten basis of
Doubilet and Rota can also be obtained by similar techniques.
omer@cs.ucsb.edu